Algebraicity of analytic maps to a hyperbolic variety

TL;DR

This paper introduces the concept of Borel hyperbolicity for algebraic varieties over complex numbers and proves its equivalence to algebraicity of maps from smooth affine curves, linking it to other hyperbolicity notions.

Contribution

It establishes a characterization of Borel hyperbolicity via holomorphic maps from curves and connects it with existing hyperbolicity concepts.

Findings

Borel hyperbolicity is equivalent to algebraicity of maps from smooth affine curves.

Borel hyperbolicity shares features with Kobayashi hyperbolicity.

The paper introduces a transcendental specialization technique for the proof.

Abstract

Let $X$ be an algebraic variety over $\mathbb{C}$. We say that $X$ is Borel hyperbolic if, for every finite type reduced scheme $S$ over $\mathbb{C}$, every holomorphic map $S^{an}\to X^{an}$ is algebraic. We use a transcendental specialization technique to prove that $X$ is Borel hyperbolic if and only if, for every smooth affine curve $C$ over $\mathbb{C}$, every holomorphic map $C^{an}\to X^{an}$ is algebraic. We use the latter result to prove that Borel hyperbolicity shares many common features with other notions of hyperbolicity such as Kobayashi hyperbolicity.